Uncoupled representation

The main conclusion of this section is that the matrix elements of all terms in the collision Hamiltonian in the fully uncoupled space-fixed representation can be reduced to simple products of integrals of the type (8.46). Such matrix elements are very easy to evaluate numerically. The fiilly uncoupled representation is therefore very convenient for the development of the coupled channel codes for collision problems involving open-shell molecules with many angular momenta that need to be accounted for. The price for simplicity is a very large number of basis states that need to be included in the expansion of the eigenstates of the full Hamiltonian to achieve full basis set convergence (see Section 8.3.4). 

As in field-free calculations, there are very many basis sets that may be used in different circumstances. A totally uncoupled representation is often convenient because the matrix elements needed are usually quite simple. For example, for collisions of a molecule in a state with an unstructured atom we need basis functions to handle the molecular rotation n, electron spin s, and end-over-end angular momentum... 

L. In this case the totally uncoupled representation is made up of simple product functions... 

The pictorial vector models of the uncoupled and coupled representations (Fig. E.l) embody the physical consequences of the angular momentum commutation rules. In the uncoupled representation, the vectors Jj and J2 can lie anywhere on their respective cones with their tips on the edges of the cones. Since these cones are invariant to rotations about the z axis, they represent... 

Figure E.l Vector models for the uncoupled representation (left) and the coupled representation (right). J , J. and are constants of the motion in the...

Such power levels can be quite adequate for Doppler-free saturation spectroscopy, as demonstrated in a series of recent studies of ultraviolet transitions of neutral helium in our laboratory at Stanford. (35-37) Fig. 5 shows as an example a spectrum of the 2 S - 5 P transition of He near 294.5 nm, recorded by intermodulated fluorescence spectroscopy. The ultraviolet radiation was generated by a yellow cw ring cavity dye laser with cavity-enhanced external ADA (ammonium dihydrogen arsenate) frequency doubler. The absorbing metastable He atoms were produced by electron impact excitation of He gas at about 0.04 torr. The spectrum shows a cluster of resolved line components which could be assigned after the fine and hyperfine Hamiltonian had been diagonalized in an uncoupled representation. We were surprised to learn that the hyperfine structure of the 5 P state of this simple 3-body system had been neither measured nor calculated before. 

Next, at point s 2. convert from the uncoupled representation back into the coupled representation according to... 

Pauli s principle is included since 4 (a,B,..., v) is zero if any pair of the sets of four quantum numbers are equal, e.g. a = 3. The Slater determinantal wavefunctions for an uncoupled representation where the good quantum numbers... 

We consider first the case of atoms whose emission spectrum can be ascribed to a single electron. This includes all the hydrogenic systems, the alkalis and the singly-ionized alkaline earth elements, etc. For the sake of simplicity at this stage we use the uncoupled representation of equation (3.61) ... 

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